Show that, in a uniform gravitational field, the net gravitational torque on a system of particles, about its centre of mass, is zero.

Let $S$ be an inertial frame of reference, and let $S^{\prime}$ be the frame of reference with the same origin and rotating with angular velocity $\boldsymbol{\omega}(t)$ with respect to $S$. You may assume that the rates of change of a vector $v$ observed in the two frames are related by

$$\left(\frac{d \mathbf{v}}{d t}\right)_{S}=\left(\frac{d \mathbf{v}}{d t}\right)_{S^{\prime}}+\omega \times \mathbf{v} .$$

Derive Euler's equations for the torque-free motion of a rigid body.

Show that the general torque-free motion of a symmetric top involves precession of the angular-velocity vector about the symmetry axis of the body. Determine how the direction and rate of precession depend on the moments of inertia of the body and its angular velocity.